Thank you all for the hints and links to many interesting materials! Well, I've understood, that I need to learn more about constructive logic and natural deduction. That book was not too detailed on these topics and authors write only strictly necessary info and from the wording I understand, they assume the readers are familiar with natural deduction at least (but I'm not). So now I'm reading some lectures on these topics and then will try the proof again. Thanks for the help!